Generalized ordered weighted aggregation robustness: a new robust counterpart to solve uncertain multi-objective optimization problems

Abstract

Based on a given set of uncertain scenarios, robust optimization aims to determine the best possible choices. The best solution in robust optimization is usually found by identifying the best out of the worst-case analysis or by using the min–max counterpart. The min–max approach is a pessimistic approach and commonly discards the effect of some uncertain scenarios in the process of finding an optimum solution. Thus, in this article, a new robust counterpart is introduced with the help of the generalized ordered weighted aggregation (GOWA) operator to solve uncertain multi-objective optimization problems with an uncertainty set of finite cardinality. After introducing GOWA robustness, the elementary properties of the GOWA robust objective function, such as continuity and local Lipschitz continuity, are analysed. Then, the relationship is discussed between the concept of GOWA robustness and (the pessimistic) min–max and (the optimistic) min–min robustness. It is shown that min–min and min–max robustness are particular cases of GOWA robustness. The GOWA robust counterpart is found to be an optimization problem with a set-valued objective mapping. Two approaches are provided to find robust solutions for GOWA: weighted-sum and epsilon-constraint. Finally, an objective-wise GOWA robust counterpart is provided to solve uncertain multi-objective optimization problems and relate it to min–max robustness. Several geometrical and numerical illustrations support the entire article.